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To solve this problem, let's break it down into steps.

### Problem Summary
- **Plane 1**: Travels at 170 km/h in a direction of \(250^\circ\).
- **Plane 2**: Travels at 150 km/h in a direction of \(200^\circ\).
- **Time**: After 4 hours, we need to find the distance between the two planes.

### Step-by-Step Solution

1. **Calculate the Distance Each Plane Travels**  
   The distance \( d \) traveled by each plane can be found with:
   \[
   d = \text{speed} \times \text{time}
   \]
   - **Plane 1**: \( d_1 = 170 \, \text{km/h} \times 4 \, \text{hours} = 680 \, \text{km} \)
   - **Plane 2**: \( d_2 = 150 \, \text{km/h} \times 4 \, \text{hours} = 600 \, \text{km} \)

2. **Convert the Directions to Cartesian Coordinates**  
   To determine how far apart the planes are, we’ll use vector coordinates.
   - **Plane 1 Position (after 4 hours)**:
     - \( x_1 = 680 \cos(250^\circ) \)
     - \( y_1 = 680 \sin(250^\circ) \)
   - **Plane 2 Position (after 4 hours)**:
     - \( x_2 = 600 \cos(200^\circ) \)
     - \( y_2 = 600 \sin(200^\circ) \)

3. **Calculate the Coordinates**
   - **For Plane 1** (\(250^\circ\)):
     - \( x_1 = 680 \cos(250^\circ) \approx 680 \times -0.3420 = -232.56 \, \text{km} \)
     - \( y_1 = 680 \sin(250^\circ) \approx 680 \times -0.9397 = -639.996 \, \text{km} \)
   - **For Plane 2** (\(200^\circ\)):
     - \( x_2 = 600 \cos(200^\circ) \approx 600 \times -0.9397 = -563.82 \, \text{km} \)
     - \( y_2 = 600 \sin(200^\circ) \approx 600 \times -0.3420 = -205.2 \, \text{km} \)

4. **Calculate the Distance Between the Two Planes**  
   Use the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\):
   \[
   \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]
   Substituting the values:
   \[
   \text{Distance} = \sqrt{(-563.82 + 232.56)^2 + (-205.2 + 639.996)^2}
   \]
   Simplify:
   \[
   = \sqrt{(-331.26)^2 + (434.796)^2}
   \]
   \[
   = \sqrt{109732.3876 + 189042.724}
   \]
   \[
   = \sqrt{298775.1116} \approx 546.64 \, \text{km}
   \]

### Final Answer
After 4 hours, the planes are approximately **547 km** apart (rounded to the nearest integer).

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Would you like further details on any part of the solution, or do you have any additional questions?

### Related Questions
1. How would the distance change if the speeds of the planes were different?
2. What if the planes flew at the same speed but in different directions?
3. How can we calculate the angles between directions using a different reference point?
4. What impact would different time intervals have on the distance calculation?
5. How do trigonometric functions apply in vector-based distance problems?

#### Tip
When solving similar problems, it’s often helpful to sketch a diagram to visualize the directions and distances.

Two airplanes leave an airport at the same time. The first flies 170 km/h in a direction of 250°. The second flies 150 km/h in a direction of 200°. After 4 hours, how far apart are the planes?

Find out how to calculate the distance between two airplanes flying in different directions at specified speeds and angles. This problem uses trigonometric concepts, vector coordinates, and distance formulas. Suitable for high school students in Grades 10-12.

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